美术馆(měi shù guǎn) - art gallery in Mandarin

美术馆 (měishùguǎn) is an art gallery, specifically one for visual art such as painting and sculpture. 美术 (měishù) means "fine art" (literally "beautiful skill/technique") and 馆 (guǎn) is the same "hall, building" character used in 图书馆 (túshūguǎn), "library", and 体育馆 (tǐyùguǎn), "sports centre".

Keep it apart from 博物馆 (bówùguǎn), "museum", which covers history, science and general collections rather than specifically art; a city typically has both, and they are not interchangeable. 画 (huà), "painting, to paint", and 艺术 (yìshù), "art" in the broader sense, both turn up in the same conversations.

去美术馆看画展 (qù měishùguǎn kàn huàzhǎn), "go to the art gallery to see an exhibition", is the phrase you will use most; a single gallery is counted with 座 (zuò), the measure word for large fixed structures, or 家 (jiā) for it as an institution.